Discrete dynamics lab: Tools for investigating cellular automata and discrete dynamical networks

Abstract

Networks of sparsely interconnected elements with discrete values and updating in parallel are central to a wide range of natural and artificial phenomena drawn from many areas of science; from physics to biology to cognition; to social and economic organization; to parallel computation and artificial life; to complex systems in general. "Decision-making" networks like this are applied as idealized models in the study of complexity and emergence, and in the behavior of networks in general, including biomolecular networks such as neural and genetic networks [3,4,6,10,12]. The networks themselves have intrinsic interest as mathematical, physical, dynamical, and computational systems with a large body of literature devoted to their study [1, 7, 8]. Because the dynamics is difficult to describe by classical mathematics, computer simulation is required, and there is a need for simulation software for nonexperts in programming to model networks in their particular fields. Discrete Dynamics Lab (DDLab) is able to construct these networks (Fig. 11.1) and investigate many aspects of their dynamical behavior. DDLab is interactive graphics software, widely used in research and education, for studying cellular automata (CA), random Boolean networks (RBN) [4], and discrete dynamical networks in general (DDN), where the "Boolean" attribute is extended to multivalue. There are currently versions of DDLab for Mac, Linux, Unix, Irix, and DOS. The source code is written in C. It may be made available on request, subject to various conditions. As well as generating space-time patterns in one, two, or three dimensions, DDLab is able to construct attractor basins, graphs that link network states according to their transitions (Fig. 11.2), analogous to Poincaré's "phase portrait" that provided powerful insights in continuous dynamics. A key insight is that the dynamics on the networks converges, thus fall into a number of basins of attraction. This is the network's memory, its ability to hierarchically categorize its patterns of activation (state-space), as a function of the precise network architecture [10]. (Figure presented) Relating this to space-time patterns in CA, high convergence implies order, low convergence implies disorder or chaos [8]. The most interesting emergent structures occur at the transition, sometimes called the "edge of chaos" [5,13]. DDLab has recently been generalized for multivalue logic. Up to eight values (or colors) are now possible, instead of just Boolean logic (two values -0,1). Of course, with just two values selected, DDLab behaves as before [15]. Multivalues open up new possibilities for dynamical behavior and modeling. Another major update is an option to constrain DDLab to run forwardonly, to generate space-time patterns for various types of totalistic rules, reducing memory load by cutting out all basin of attraction functions. This allows larger neighborhoods (max-k=25, instead of 13). In 2D the neighborhoods are predefined to make hexagonal as well square lattices. Many interesting cellular automaton rules with "life"-like and other complex dynamics can be found in totalistic multivalue rule-space, in 3D as well as 2D [16]. DDLab is an applications program, it does not require writing code. Network parameters and the graphics presentation can be flexibly set, reviewed and altered interactively, including changes on-the-fly. There are built-in tools for constructing and manipulating networks. A wide variety of measures, data, analysis, and statistics are available. For small networks, it is possible to compute and draw basins of attraction, and measure their convergence and stability to perturbation. For larger networks, basins of attraction can be investigated statistically. This chapter provides some general background and gives the flavor of DDLab with a range of examples; the figures shown were all produced within DDLab. The operating manual [14] describes all of DDLab's many functions and includes a "quick start" chapter. DDLab is available at www.ddlab.org and www.cogs.susx.ac.uk/users/andywu/ddlab.html. DDLab remains free shareware for personal, noncommercial, users. Any other users, including commercial users, companies, government agencies, research or educational institutions, must register and pay a license fee (see www.ddlab.org/ddinc.html) . © Springer-Verlag London Limited 2005.